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# Show that the function defined by g(x) = x – [x] is discontinuous at all integral points. Here [x] denotes the greatest integer less than or equal to x.

19. Show that the function defined by $g (x) = x- [x]$ is discontinuous at all integral points. Here [x] denotes the greatest integer less than or equal to x.

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Given function is
$g (x) = x- [x]$
Given is defined for all real numbers k
$\lim_{x\rightarrow k^-}f(x) = k - (k-1) = k-k+1 =1\\ \lim_{x\rightarrow k^+}f(x) = k - k = 0\\ \lim_{x\rightarrow k^-}f(x) \neq \lim_{x\rightarrow k^+}f(x)$
Hence, by this, we can say that  the function defined by $g (x) = x- [x]$ is discontinuous at all integral points

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